Transcript Chapter 11 - Cengage Learning

Chapter 11: Probability
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Objectives
•
•
•
•
•
•
Define probability
Work with mutually exclusive events
Work with independent events
Work with dependent events
Calculate expected monetary values
Recognise and calculate binomial
probabilities
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Defining Probability
Most commonly used definition:
P(Event) =
Number of ways event can occur
Number of outcomes
For example, a coin has 2 sides,
so probability of a Head is 1/2
BUT
The problem is that not all outcomes are
necessarily equally likely
or equally probable
(Think of a biased dice)
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
More Definitions
Frequency definition:
Number of times event occurs
P(Event) =
Number of trials
Subjective definition:
Ask a series of “experts” to estimate the probability
and use feedback to refine the figure
Axiomatic approach:
Set up axioms and derive probability results from these
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Mutually Exclusive Events
If two (or more) events cannot happen at the same time,
they are called mutually exclusive.
This could be different scores on a die,
since only one face can show at a time.
In this case we can just add the probabilities together:
P(1) + P(2) = 1/6 + 1/6 = 1/3 = P(1 or 2)
Rule 1: Where A and B are mutually exclusive, then:
P(A or B) = P(A) + P(B)
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Non-Mutually Exclusive Events
Some items or events have more than one characteristic
If you think of a pack of cards, each card has a value and a suit.
Sample space:
A Clubs
K
Hearts
K
Diamonds
K
Spades
Rule 2: Where A and B are not mutually
exclusive, then:
P(A or B) = P(A) + P(B) – P(A and B)
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
K Clubs
Q Clubs
J Clubs
10 Clubs
9 Clubs
8 Clubs
7 Clubs
7 Clubs
6 Clubs
5 Clubs
4 Clubs
3 Clubs
2 Clubs
Independent Events
If the outcome of one event does not affect the outcome of
some other event, then the 2 events are independent
For example, if two coins are tossed, the result on the first
coin has no effect on the result on the second coin
So we can multiply the probabilities
P(H on 1st) x P(H on 2nd) = ½ x ½=¼ = P(H on both
coins)
Rule 3 Where A and B are independent, then:
P(A & B) = P(A) x P(B)
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Dependent Events
Where the outcome of one event affects the outcome of a second
event, the second event is called dependent
For example, if we have a small group of people consisting of 3
men and 4 women and want to find probabilities relating to
selecting 2 people, then the probability relating to the second
person will depend on who was selected first.
First Person
P(Man) = 3/7
P(Woman) = 4/7
2nd person if 1st
Male
P(Man) = 2/6
P(Woman) = 4/7
2nd person if 1st
Female
P(Man) = 3/6
P(Woman) = 3/7
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Expected Values
Probability is a useful concept, but many people find it a little
difficult, unless they are doing a course such as this.
They will want to know what the effect or outcome is, and we can
help by talking about what is expected to happen
This expectation is a sort of “average in the long run”
And we find it by multiplying the effect by the probability
So, if we have a lottery ticket which might win £1,000 with a
probability of 0.001, but win nothing otherwise, then we have the
expected value of the ticket as:
£1,000x0.001 + £0x0.999 = £1
Don’t think this is what will happen – it can’t!
It is the average value of the ticket, so don’t pay £2 for it!
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Decision Trees
This is a way of illustrating the consequences of decisions
and the outcomes of events for which we can assign
probabilities
As a visual medium, it can help communicate ideas to other
people
It can also help you to think through a problem, making sure
that you have considered every option.
When using decision trees we are usually dealing with
monetary outcomes, so rather than expected values, we
usually talk about
Expected Monetary Values
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Decision Trees (2)
Think about the following example:
A company is deciding whether or not to expand.
If they expand, at a cost of £50,000 and the market also
expands, their expected increase in profit is £100,000 per
year. Expansion without an increase in the market might
give increased profits of £20,000 per year.
If they do not expand and the market grows, they will still
increase their profits by £20,000 per year, but if the market
does not grow, then there is no foreseeable increase in
profits.
The probability the market grows as predicted is 0.25
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Decision Trees (3)
For the expand company option, we have:
Expected Profit
Market expands
p=0.25
Market doesn’t expand
p=0.75
£100,000*.25
=£25,000
£20,000*.75
=£15,000
The overall expected profit increase is £25,000 + £15,000
= £40,000
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Decision Trees (4)
For the do not expand company option, we have:
Expected Profit
Market expands
p=0.25
Market doesn’t expand
p=0.75
£20,000*.25
=£5,000
£0*.75
=£0
The overall expected profit increase is £5,000 + £0
= £5,000
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Decision Trees (5)
Putting the two diagrams together, we get:
Market grows
Expected Profit
£25,000
Expand
Cost=£50,000
No growth
Market grows
Don’t expand
Cost = £0
No growth
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
£15,000
£5,000
£0
Decision Trees (6)
Looking at the overall outcomes
If the company expands the cost is £50,000 and the
expected increase in profit is £40,000
a net expected cost of £10,000
If the company does not expand, the cost is £0 and the
expected increase in profit is £5,000
a net expected profit of £5,000
You would recommend the company not to expand
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Binomial Probabilities
A binomial model will apply where
(a) the chance of success (however defined) is constant
from event to event
(b) there are two outcomes (or at least, the outcomes
can be classified into just two) – success and failure
Binomial models have proved useful in solving many
problems and are fairly easy to use.
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Binomial Results
If there are two trials, where the probability of success is
0.4, then there are 4 possible results:
Result
Two successes.
Calculation
0.4 x 0.4
0.4 x 0.6
Probability
0.16
Failure on the first trial and
success on the second
0.6 x 0.4
0.24
Two failures
0.6 x 0.6
0.36
Success on the first trial
and failure on the second.
Total
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
0.24
1.00
Combining Results
Looking at the last table, you can see that the middle two
results are the same, except for the order in which the
events occur
Since order doesn’t matter, we can combine these to get:
Result
Two successes.
One success
Calculation
0.4 x 0.4
2 x 0.4 x 0.6
Zero successes
0.6 x 0.6
Total
Probability
0.16
0.48
0.36
1.00
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Binomial Formula
We can derive a general formula for binomial situations
where n = number of trials
and p = probability of success
and r = number of successes
 n r
nr
P(r successes in n trials )    p 1  p 
r 
The number of different ways of getting r successes is given by:
 n
n!
  
 r  r!(n  r )!
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Binomial Example
If we pick 4 items at random from a production process where
we know that the probability of a faulty item is constant at 0.1
What is the probability of getting 2 faulty items in the sample?
In this case we can identify that p = 0.1, n = 5 and r = 2
Putting these into our formula gives:
 5  5! 5.4.3! 20
  


 10
2
 2  2!3! 2.3!
The probability will be:
P(2)  10(0.1) 2 (0.9) 3  10  0.01  0.729  0.0729
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004
Conclusions
•
•
•
Probability provides a very useful means of
looking at problems.
It highlights the fact that little is absolutely
certain and that alternative outcomes have
some chance of happening.
Learning to think using probability will help
you to consider areas such as contingency
planning and risk analysis.
Jon
Jon Curwin
Curwin and
and Roger
Roger Slater,
Slater, QUANTITATIVE
QUANTITATIVE METHODS:
METHODS: A
A SHORT
SHORT COURSE
COURSE
ISBN 1-86152-991-0
ISBN 1-86152-991-0
© Thomson
© Cengage
Learning 2004